1. Field of the Invention
The present invention relates to a method of simulating useful in analyzing, with good accuracy, deformation and the like of the viscoelastic material.
2. Description of the Related Art
The viscoelastic material as represented by rubber is widely used in for example, tires and industrial goods such as sporting goods. The viscoelastic material deforms greatly when subjected to load, and restores to the original state when the load is completely removed or is unloaded. The viscoelastic material has a non-linear elastic behavior under static load and a rate-dependent or viscoelastic behavior with hysteresis under cyclic loading. To reduce the trouble and the cost of experimental manufacture, a simulation of for example, deformation process of the viscoelastic material is carried out using a computer. A conventional simulation method of the viscoelastic material is disclosed in Japanese Laid-Open Patent Publication No. 2002-365205.
The above mentioned publication focuses on the fact that the viscoelastic material shows a different modulus of longitudinal elasticity in accordance with strain velocity. More specifically, strain, strain velocity and stress produced in the relevant viscoelastic material are measured under a measuring condition assuming, in advance, the actual usage state of the viscoelastic material. Thus, the corresponding relationship between the modulus of longitudinal elasticity and the strain velocity is obtained. With respect to a viscoelastic material model serving as an analyzing object, a predetermined strain velocity is given and the modulus of longitudinal elasticity is appropriately calculated from the above corresponding relationship to perform the deformation calculation.
The simulation method of the viscoelastic material is described in for example, the following article.
“A THREE-DIMENSIONAL CONSTITUTIVE MODEL FOR THE LARGE STRETCH BEHAVIOR OF RUBBER ELASTIC MATERIALS” by Ellen M. Arruda and Marry C. Boyce, Journal of the Mechanics and Physics of Solids Volume 41, Issue 2, Pages 389-412 (February 1993).
The content thereof will now be briefly described.
The above mentioned article is premised on the molecular chain network model theory in which the viscoelastic material has a network structure as a microscopic structure. As shown in FIG. 35, the network structure of the viscoelastic material “a” includes a plurality of molecular chains c linked at a linking point b. The linking point b includes a chemical linking point between the molecules such as for example, a chemical cross-linking point.
One molecular chain c is configured by a plurality of segments e. One segment e is the smallest constitutional unit for repetition. Further, one segment e is configured by joining a plurality of monomers f in which carbon atoms are linked by covalent bonding. Carbon atoms each freely rotates with respect to each other around a bond axis between the carbons. Thus, the segment e can be bent, as a whole, into various shapes.
In the above mentioned article, with respect to the fluctuation cycle of the atom, the average position of the linking point b does not change in the long term. Therefore, the perturbation about the linking point b is ignored. Further, the end-to-end vector of the molecular chain c having two linking points b, b on both ends is assumed to deform with a continuous body of the viscoelastic material “a” to which the molecular chain is embedded.
Aruuda et al. also proposes an eight chain rubber elasticity model. As shown in FIG. 6, this model is defined the macroscopic structure of the viscoelastic material as a cubic network structure body h in which the microscopic eight chain rubber elasticity models g are collected. In one eight chain rubber elasticity model g, the molecular chain c extends from one linking point b1 placed at the center of the cube to each of the eight linking point b2 at each apex of the cube, as shown enlarged on the right side of FIG. 6.
In simulation, the viscoelastic material is defined as a super-elastic body in which volume change barely occurs and in which restoration to the original shape occurs after the load is removed. The super-elastic body is, as expressed in the following equation (1), defined as a substance having a strain energy function W that is differentiated by a component Eij of Green strain to produce a conjugate Kirchhoff stress Sij. In other words, the strain energy function shows the presence of potential energy stored when the viscoelastic material deforms. Therefore, the relationship between the stress and the strain of the super-elastic body is obtained from a differential slope of the strain energy function W.
                              S          ij                =                              ∂            W                                ∂                          E              ij                                                          (        1        )            
Aruuda et al. recognized, based on the non-Gaussian statistics theory, that as the deformation of the viscoelastic material increases, the entropy change increases dramatically (the molecular chain is stretched and oriented), and thus showed the strain energy function W of a rubber elastic body expressed in equation (2). Moreover, by substituting the strain energy function W to the above mentioned equation (1), the relationship between the stress and the strain of the viscoelastic material is obtained.
                                                        W              =                                                                    C                    R                                    ⁢                                      N                    (                                                                                                                                                      I                              1                                                                                      3                              ⁢                              N                                                                                                      ·                        β                                            +                                              ln                        ⁢                                                                                                  ⁢                                                  β                                                      sinh                            ⁢                                                                                                                  ⁢                            β                                                                                                                )                                                  -                                  T                  ·                  n                  ·                  c                                                                                                                        C                R                            =                              n                ·                                  k                  B                                ·                T                                                                        (        2        )            (n: number of molecular chains per unit volume; kB: Boltzmann constant; T: absolute temperature)    I1: primary invariable quantity of strain, I1=λ12+λ22+λ32 (λ1, λ2, λ3 are elongation ratios)    N: average segment number per one molecular chain
  β  =            L              -        1              ⁡          (                        r          chain                          N          ·          a                    )      (rchain: distance between the ends of one molecular chain,
                              N          ·                      I            1                          3              ·    a    ;      a    ⁢          :      length of one segment; L: Langevin function,
            L      ⁡              (        x        )              =                  coth        ⁢                                  ⁢        x            -              1        x              )
By performing the uniaxial tensile deformation simulation of the viscoelastic material using the relationship between the stress and the strain defined by Aruuda et al., a non-linear relationship between the stress and the strain is obtained, as shown in for example, FIG. 36. This result shows a good correlation with the actual measurement during the loaded deformation.
In the viscoelastic material used as industrial goods, filler (filling agent) such as carbon black and silica is usually blended. To perform the deformation simulation of the viscoelastic material blended with filler with good accuracy, it is not appropriate to ignore the existence of the filler. However, in the conventional simulation, such filler is not taken into consideration.
As a result of a various experiments, various specific phenomenon such as slippage and friction of the matrix and the filler occurs at the interface of the filler and the matrix, and a relatively large energy loss occurs at such region. Therefore, to carry out the simulation of the viscoelastic material with a good accuracy, it is important to take such phenomenon at the interface into consideration in the calculation.